Sigma Notation enables to write a sum of many terms in a sequence in compact form. We have used the sigma notation to represent the sum of a finite number of terms in arithmetic and geometric sequences. The Greek letter ∑ corresponds to English letter S, hence used as a notation to represent sum.

The simplest form of sigma notation consists of the letter ∑ with boundary values written on top and bottom and the general term of the sequence is written inside the notation. This when expanded shows all the terms of the sequence summed up as shown below.
$\sum_{i=1}^{n}a_{i}$ = a1 + a2 +.......+ an.

The Syntax of Sigma Notation is shown as follows:

Sigma Notation

The letter i is called the index of the Sigma Notation. It is used as a suffix to represent the general term of Series. The lower and higher boundary values of i are written correspondingly at the bottom and top of the Greek Letter Sigma.

For example, the sum to n terms of a Geometric Sequence with first term = a and common difference r is written as
$\sum_{n=1}^{m}$ arn-1.

Suppose a = 1 and r = 3, the sum to 10 terms of the Geometric Sequence is represented using Sigma Notation as
$\sum_{n=1}^{10}$ 3n-1.
The following algebraic rules that allow working with Sigma Notations can be considered as Properties of Sigma Notation. These rules are applicable only with finite sums.
  1. Sum Rule: $\sum_{k=1}^{n}$ (ak + bk) = $\sum_{k=1}^{n}$ ak + $\sum_{k=1}^{n}$ bk.
  2. Difference Rule: $\sum_{k=1}^{n}$ (ak - bk) = $\sum_{k=1}^{n}$ ak - $\sum_{k=1}^{n}$ bk.
  3. Constant Multiple Rule: $\sum_{k=1}^{n}$ cak = c.$\sum_{k=1}^{n}$ak.
  4. Constant Value Rule: $\sum_{k=1}^{n}$ c = n.c

In addition to these, the formulas for finding some special sums are related to Sigma Notation.

  1. $\sum_{k=1}^{n}$ k = $\frac{n(n+1)}{2}$ Sum of the first n positive integers
  2. $\sum_{k=1}^{n}$ k2 = $\frac{n(n+1)(2n+1)}{6}$ Sum of the squares of the first n positive integers
  3. $\sum_{k=1}^{n}$ k3 = $\left [ \frac{n(n+1)}{2} \right ]^{2}$ Sum of the cubes of the first n positive integers

Let us look at some example problems where the properties of sigma notation are used to solve.

Compute $\sum_{n=1}^{10}$ (3n + 4)
$\sum_{n=1}^{10}$ (3n + 4) = $\sum_{n=1}^{10}$ 3n + $\sum_{n=1}^{10}$

4. (Sum Rule for Sigma Notation)
= 3 $\sum_{n=1}^{10}$ n + 4 x 10 (Constant multiple Rule and Constant value Rule)
= 3 x $\frac{10\times 11}{2}$ + 40 (Sum of the first 10 positive integers)
= 165 + 40

= 205

Riemann sum is used to estimate the area under the curve, by partitioning the area into a finite number of rectangles of equal widths.
The formula used to find the Riemann sum is given in the form of Sigma Notation for finite sums.

A ≈ $\sum_{i=1}^{n}$ f(xi) Δx Where xi is a point in the ith interval.
Definite integral in a given interval [a, b] is defined as the limit of this sum as n → ∞. The definite integral of f(x) from a to b is defined using infinite sigma notation as follows:
$\int_{a}^{b}f(x)dx$ = $\lim_{n\rightarrow \infty }$ $\sum_{i=1}^{n}$ f(xi) Δx

Infinite Sigma notation is used to represent the sum of infinite sequence. If the general term of the sequence is represented by an, then the sum of the sequence is called the infinite series and represented infinite sigma notation $\sum_{n=1}^{\infty }$ an.

The following equation relates the series condensed in infinite sigma notation to the expanded form.
$\sum_{n=1}^{\infty }$ an = a1 + a2 + a3 +................

Example:
$\sum_{n=1}^{\infty }$ $\frac{1}{3^{n}}$ = $\frac{1}{3}$ + $\frac{1}{9}$ + $\frac{1}{27}$ + $\frac{1}{81}$ + .......

Solved Examples

Question 1: Expand the Sigma notation and evaluate:
$\sum_{k=1}^{3}$ $\frac{k}{k+1}$
Solution:
 
The sigma notation represents a finite sum of three terms. Substituting k = 1, 2 and 3 we get

$\sum_{k=1}^{3}$ $\frac{k}{k+1}$ = $\frac{1}{1+1}$ + $\frac{2}{2+1}$ + $\frac{3}{3+1}$

                                                = $\frac{1}{2}$ + $\frac{2}{3}$ + $\frac{3}{4}$ = $\frac{23}{12}$.

In the above example we found the sum of the series with only three terms. But the terms can also be expanded in the middle of a series and the required sum found.
 

Question 2: Find the sum of the 5th, 6th and the 7th terms in the infinite series $\sum_{n=1}^{\infty }$ $\frac{n}{n!}$
Solution:
 
The sum of the 5, 6 and the 7 terms can be represented using sigma notation as

$\sum_{n=5}^{7}$ $\frac{n}{n!}$ = $\frac{5}{5!}$ + $\frac{6}{6!}$ + $\frac{7}{7!}$

                                             = $\frac{1}{24}$ + $\frac{1}{120}$ + $\frac{1}{720}$ = $\frac{37}{720}$
 

Question 3: Use the properties of sigma notation to compute the sum $\sum_{i=1}^{10}$ i3 + i2 + i.
Solution:
 
Using the sum rule for sigma notation the sum can be separated as,

$\sum_{i=1}^{10}$ i3 + i2 + i = $\sum_{i=1}^{10}$ i3 + $\sum_{i=1}^{10}$ i2 + $\sum_{i=1}^{10}$ i
Use the sum formulas substituting n =10

$\sum_{i=1}^{10}$ i3 = $\left [ \frac{i(i+1)}{2} \right ]^{2}$ = $\left [ \frac{10\times 11}{2} \right ]^{2}$ = 552 = 3025

$\sum_{i=1}^{10}$ i2 = $\frac{i(i+1)(2i+1)}{6}$ = $\frac{10\times 11\times 21}{6}$ = 385

$\sum_{i=1}^{10}$ i = $\frac{i(i+1)}{2}$ = $\frac{10\times 11}{2}$ = 55

Hence
$\sum_{i=1}^{10}$ i3 + i2 + i = $\sum_{i=1}^{10}$ i3 + $\sum_{i=1}^{10}$ i2 + $\sum_{i=1}^{10}$ i
                                         = 3025 + 385 + 55 = 3465

 

Question 4: Express the sum 2 - 4 + 8 - 16 + 20 - .........$\infty$ in sigma notation.
Solution:
 
You may note the signs alternate in the above series. The odd numbered terms are positive while the even terms are negative in sign. Observing the pattern of numbers, the number in the nth term = 2n. Thus the above alternating infinite series can be expressed in sigma notation as
$\sum_{i=1}^{\infty }$ (-1)n-1 2n.
 

Expand and find the sum of

Practice Problems

Question 1: $\sum_{k=1}^{6}$ 2k-1.                           (Ans: 63)
Question 2: $\sum_{n=0}^{5}$ cos(nπ)                      (Ans: 0)
Question 3: Find the sum using the properties of sigma notation and the formulas for sum. $\sum_{i=1}^{100}$ (i3 - 2i2)                    (Ans: 24,825,800)